Method for estimating the concentration of a tracer in a tissue structure assembly, and corresponding storage medium and device

ABSTRACT

A method provides for estimating the concentration of a tracer in a tissue structure assembly including at least one tissue structure, from a measurement image of the tracer concentration in said tissue structure assembly, which is obtained by an imaging apparatus, wherein said image includes at least one space domain inside which the tracer concentration is homogenous and at least one region of interest in which the tracer concentration is measured. The method includes: determining a geometric transfer matrix having coefficients representative of the contribution of the space domains in the measurement of the tracer concentration in the regions of interest; optimizing the coefficient of the geometric transfer matrix by defining the best regions of interest in terms of errors in order to measure the tracer concentration, the definition of the regions of interest being carried out according to an iterative loop that includes the following steps upon each iteration: modifying the regions of interest, and calculating the coefficients of the geometric transfer matrix from the modified regions of interest; selecting an optimized geometric transfer matrix among the calculated geometric transfer matrices; and estimating the tracer concentration from the optimized geometric transfer matrix.

FIELD OF THE INVENTION

The present invention relates to a method for estimating the concentration of a tracer in a tissue structure assembly comprising at least one tissue structure, from a measurement image of the tracer concentration in said tissue structure assembly, which is obtained by an imaging apparatus, the image comprising at least one space domain inside which the tracer concentration is homogenous and at least one region of interest inside which the tracer concentration is measured.

BACKGROUND

In vivo molecular imaging makes it possible to measure local biochemical and pharmacological parameters non-invasively in intact animals and humans. Molecular imaging includes nuclear techniques, magnetic resonance imaging (MRI), and optical techniques. Among nuclear techniques, positron emission tomography (PET) is the most sensitive technique and the only one to offer quantitative measurements.

The quantification of measurements with PET for the study of a drug requires the intravenous injection of a radioactive tracer adapted to the study of that drug. An image of the tracer distribution is then reconstructed. The time activity curves (TAC), i.e. the variations of the tracer concentration within a volume element, are also highly spatially correlated due not only to the reconstruction method, but also to the existence of physiological regions that respond to the tracer identically, regions that can be called pharmaco-organs.

The post-processing of the PET image generally requires three steps to access the pharmacological parameter. First, regions of interest (ROI) defining the organs, more precisely the pharmacokinetic organs, must be defined either on the PET image or on a high-resolution image matched point-to-point with the PET image. Next, the TACs of the pharmaco-organs must be extracted and possibly corrected to offset the limited spatial resolution of the PET. Lastly, a physiological model can be defined, based on the TACs and the tracer concentration in the plasma, making it possible to calculate pharmacological parameters that are interesting for the biologist. A precise definition of the ROIs is necessary so as to extract the relevant TACs.

However, the quantification of the TACs is hindered by the limited resolution of the PET system, resulting in what is called the partial volume effect (PVE).

A known geometric transfer matrix (GTM) method makes it possible to correct the PVE effectively. This method requires the data from the PET image on one hand, and on the other hand, space domains defining the functional organs, regions of interest within which the average TACs of the functional organs will be calculated, and a resolution model of the PET imager.

However, this GTM method is very sensitive not only to the definition errors of said space domains and the image reconstruction artifacts, but also to the image smoothing effects due to periodic physiological movements, such as heartbeats or respiratory movements.

SUMMARY

The present invention aims to propose a method making it possible to limit the impact of the effects due to segmentation errors, image reconstruction artifacts, and physiological movements on the effectiveness of the GTM method in order to obtain TACs having the smallest possible bias with the lowest uncertainty due to noise.

To that end, the present invention relates to a method of the aforementioned type, wherein the method comprises the following steps:

-   -   determining a geometric transfer matrix having coefficients         representative of the contribution of the space domains in the         measurement of the tracer concentration in the regions of         interest;     -   optimizing the coefficients of the geometric transfer matrix by         defining the best regions of interest in terms of errors in         order to measure the tracer concentration, the definition of the         regions of interest being carried out according to an iterative         loop that includes the following steps upon each iteration:         -   modifying the regions of interest; and         -   calculating the coefficients of the geometric transfer             matrix from the modified regions of interest;     -   selecting an optimized geometric transfer matrix among the         calculated geometric transfer matrices; and     -   estimating the tracer concentration from the optimized geometric         transfer matrix.

The method according to the present invention may include one or more of the following features:

-   -   the step for modifying the regions of interest comprises the         addition and/or exclusion of at least one image element;     -   the iterative loop for defining the regions of interest         comprises, for each iteration, the following steps:         -   determining a set of elementary modifications each including             or consisting of adding at least one image element to the             regions of interest;         -   estimating, for each elementary modification, the tracer             concentration in the regions of interest modified by the             elementary modification;         -   evaluating, for each elementary modification, a criterion             making it possible to compare the estimated tracer             concentration with the tracer concentrations estimated in at             least one preceding iteration;         -   selecting the elementary modification yielding a tracer             concentration estimate closest to the tracer concentrations             estimated in the preceding iterations;         -   comparing the criterion corresponding to the selected             elementary modification with a predetermined threshold; and         -   applying or not applying the selected elementary             modification or stopping the iterative loop as a function of             the result of the comparison of the criterion with the             threshold;     -   the method comprises a step for ordering image elements in each         space domain as a function of order criteria making it possible         to evaluate their risk of introducing errors not due to noise         into the tracer concentration estimate;     -   the order criteria comprise at least one from amongst a first         criterion making it possible to define whether the concentration         contained in the image element comes from one or more different         spatial domains, a second criterion making it possible to define         whether the image element participates in the calculation of the         non-diagonal elements of the geometric transfer matrix, and a         third criterion making it possible to define whether the image         element is reliable for estimating the tracer concentration;     -   the first criterion makes it possible to measure the homogeneity         of the tracer concentration near the image element, the second         criterion makes it possible to measure the contribution of the         space domains to the measurement of the tracer concentration in         the image element, and the third criterion comes from a         probabilistic atlas;     -   the iterative loop for defining the regions of interest is         carried out following the order of the image elements obtained         in the ordering step;     -   the method comprises a step for estimating the point spread         function of the imaging apparatus at any point of the imaging         apparatus's field of view;     -   the step for estimating the point spread function comprises the         following steps:         -   measuring the point spread function at several points of the             field of view; and         -   interpolating and/or extrapolating the obtained             measurements;     -   the method comprises a step for estimating a region spread         function for each space domain by convolution of the point         spread function with each space domain;     -   the method comprises a step for calculating the size of the         regions of interest;     -   the method comprises a step for delimiting space domains so as         to obtain regions of interest with a maximum size;     -   the method comprises a step for optimizing order criteria so as         to obtain regions of interest with a maximum size;     -   the measurement image of the tracer concentration in the tissue         structure is a sequence of matched three-dimensional images and         comprising at least one three-dimensional image, and the image         elements are voxels; and     -   the three-dimensional image is obtained by positron emission         tomography.

The present invention also relates to a storage medium comprising a code to estimate the concentration of a tracer in a tissue structure assembly comprising at least one tissue structure from a measurement image of the tracer concentration in said tissue structure assembly obtained by an imaging apparatus, the image comprising at least one space domain inside which the tracer concentration is homogenous and at least one region of interest inside which the tracer concentration is measured, wherein the code comprises instructions to:

-   -   determine a geometric transfer matrix having coefficients         representative of the contribution of the space domains in the         measurement of the tracer concentration in the regions of         interest;     -   optimize the coefficients of the geometric transfer matrix by         defining the best regions of interest in terms of errors in         order to measure the tracer concentration, the definition of the         regions of interest being carried out according to an iterative         loop that includes the following steps upon each iteration:         -   modifying the regions of interest; and         -   calculating the coefficients of the geometric transfer             matrix from the modified regions of interest;     -   select an optimized geometric transfer matrix among the         calculated geometric transfer matrices; and     -   estimate the tracer concentration from the optimized geometric         transfer matrix,     -   when it is executed by a data processing system.

The present invention also relates to a device intended to estimate the concentration of a tracer in a tissue structure assembly comprising at least one tissue structure from a measurement image of the tracer concentration in said tissue structure assembly obtained by an imaging apparatus, the image comprising at least one space domain inside which the tracer concentration is homogenous and at least one region of interest inside which the tracer concentration is measured, wherein the device comprises:

-   -   an imaging apparatus; and     -   a data processing system comprising:         -   means for determining a geometric transfer matrix having             coefficients representative of the contribution of the space             domains in the measurement of the tracer concentration in             the regions of interest;         -   means for optimizing the coefficients of the geometric             transfer matrix by defining the best regions of interest in             terms of errors in order to measure the tracer             concentration, the definition of the regions of interest             being carried out according to an iterative loop that             includes the following steps upon each iteration:             -   modifying the regions of interest; and             -   calculating the coefficients of the geometric transfer                 matrix from the modified regions of interest;         -   means for selecting an optimized geometric transfer matrix             among the calculated geometric transfer matrices; and         -   means for estimating the tracer concentration from the             optimized geometric transfer matrix.

The present invention will be better understood upon reading the following description, provided solely as an example and done in reference to the appended drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flowchart showing the two main phases of the inventive method.

FIG. 2 is a diagram illustrating the PET image of a mouse.

FIG. 3 is a flowchart showing the steps of the inventive method.

FIGS. 4 and 5 are respective transaxial and longitudinal cross-sections of the field of view of the PET scanner.

FIG. 6 is a flowchart showing the step for defining the regions of interest in more detail.

FIGS. 7A to 11B are diagrams showing the results obtained with one example of an embodiment of the inventive method.

DETAILED DESCRIPTION

The present invention is based on the geometric transfer matrix (GTM) method of correcting the partial volume effect (PVE) proposed by Rousset et al. in the article “Correction for Partial Volume Effects in PET: Principle and Validation”, The Journal of Nuclear Medicine, 1998, Vol. 39, No. 5, pages 904 to 911, and implemented in the image area by Frouin et al. in the article “Correction of Partial-Volume Effect for PET Striatal Imaging: Fast Implementation and Study of Robustness”, The Journal of Nuclear Medicine, 2002, Vol. 43, No. 12, pages 1715 to 1726.

The aim of the GTM method is to find the true tracer concentration T_(i) inside the space domains D_(i). The details of the GTM method can be found in the aforementioned article by Rousset et al., but the principles are presented here so as to better highlight the features of the present invention.

The method according to the present invention aims to optimize the calculation parameters of the true tracer concentration T_(i) by limiting the effects due to segmentation errors, artifacts from reconstructing the image, and physiological movements.

As shown in FIG. 1, the method according to the present invention comprises two main phases:

-   -   a first phase 10 intended to suitably select the response         function of the PET scanner; and     -   a second phase 12 intended to choose the optimal regions of         interest for estimating the true tracer concentration T_(i).

In reference to FIG. 2, the PET image is a three-dimensional image formed from a set of voxels X(x,y,z) each characterized by a tracer concentration A(x,y,z), said concentration being capable of varying over time.

The PET image is assumed to be made up of I homogenous space domains {D_(i)}_(1≦i≦I) in terms of tracer concentration. Each space domain D_(i) has a true tracer concentration denoted T_(i).

A tracer concentration {t_(j)}_(1≦j≦J) is measured in the PET image inside regions of interest (ROI) {R_(j)}_(1≦j≦J).

For clarity of the explanation, we will hereafter consider that each ROI is included in the corresponding space domain and that their numbers are equal: J=I.

Alternatively, J is different from I, preferably with J≧I so that the GTM method makes it possible to find the true concentration T_(i) of the tracer.

The measured tracer concentration t_(j) in a region of interest R_(j), with volume V_(j), can be expressed as a linear combination of the tracer concentrations {T_(j)}_(1≦≦I) in the different space domains:

$\begin{matrix} {t_{j} = {\frac{1}{V_{j}}{\sum\limits_{i = 1}^{I}{T_{i} \times {\int_{R_{j}}{{{RSF}_{i}(x)}\ {\mathbb{d}x}}}}}}} & (1) \end{matrix}$

where RSF_(i)(x) is the contribution of the space domain D_(i) to the PET measurement of the tracer concentration in voxel x.

This equation can be rewritten in matricial form: t=W·T  (2)

where t is the vector of the averaged tracer concentration PET measurements in the ROIs R_(j), T is the vector of the true tracer concentrations in the space domains D_(i), and W is the geometric transfer matrix (GTM) I×J between the space domains {D_(i)}_(1≦i≦I) and the ROIs {R_(j)}_(1≦j≦J).

The coefficients w_(i,j) of the matrix W are equal to:

$\begin{matrix} {w_{i,j} = {\frac{1}{V_{j}}{\int_{R_{j}}{{{RSF}_{i}(x)}\ {\mathbb{d}x}}}}} & (3) \end{matrix}$

The true tracer concentration can then be calculated as: T=W ⁻¹ ·t  (4)

As illustrated in FIG. 3, an initial step 14 of the phase 12 of the inventive method consists of defining the space domains D_(i).

The user is free to define the D_(i) as he wishes, by drawing them manually, semi-automatically, or automatically on the PET image, but with the constraint that each space domain D_(i) is homogenous in terms of true tracer concentration, this concentration being able to vary over time in the case where the PET image contains dynamic information.

Alternatively, the D_(i) are drawn on an image coming from another imaging form, such as nuclear MRI or X-ray scanner.

In parallel, the phase 10 for selecting the response function of the PET scanner is carried out.

The response function of the scanner, or point spread function (PSF), is preferably estimated so that the GTM method works effectively.

In reference to FIGS. 4 and 5, the PSF is broken down in a cylindrical coordinate system (ρ,θ,z) into:

-   -   an axial component along the z axis: PSF_(axial);     -   a radial component in direction CX: PSF_(radial); and     -   a tangential component in direction XT: PSF_(tangential).

The components PSF_(axial), PSF_(radial) and PSF_(tangential) of the PSF can each comprise several parameters, for example two variances of two gaussians if the PSF is estimated by two gaussians, or several values.

If we assume that the PSF is invariable along z, it is only necessary to measure the PSF on the transaxial plane passing through the middle of the PET scanner. According to this hypothesis, PSF_(axial) does not vary with the position in the field of view, and PSF_(radial) and PSF_(tangential) are also invariable along θ. It is therefore sufficient to measure PSF_(axial) at a point of the field of view, and PSF_(radial) and PSF_(tangential) at several points of the x axis.

If one assumes that the PSF varies as one moves towards the axis of the PET scanner, it is necessary to measure PSF_(axial) at several points on the z axis, and PSF_(radial) and PSF_(tangential) at several points on the x axis.

In reference to FIG. 3, a first step 16 of phase 10 of the inventive method consists of measuring PSF_(axial), PSF_(radial) and PSF_(tangential) at several points of the field of view of the PET scanner.

According to the present invention, the PSF may have any parametric or non-parametric form, or more precisely a form in line with the PSF form generated by the method for reconstructing the PET image, and possibly also with the smoothing effect carried out by the physiological movements. If the PET image is reconstructed with modeling or compensation of the PSF, the PSF in the context of the present invention will model the residual effects not taken into account during the reconstruction (for example if the reconstruction assumes the PSF to be uniform in the field of view, then the deviation for each point or region of the field of view between the uniform PSF and the actual PSF will be modeled).

In a second step 18 of phase 10, the value of PSF_(axial), PSF_(radial) and PSF_(tangential) is deduced from these measurements at all points of the field of view by interpolation and/or extrapolation, using a method selected by the user.

As previously seen, PSF_(axial) only depends on the position on the z axis and will be denoted PSF_(axial)(z), whereas PSF_(radial) and PSF_(tangential) only depend on ρ and will therefore be denoted PSF_(radial)(ρ) and PSF_(tangential)(ρ).

In step 20 of the inventive method, RSF_(i)(x) will be estimated at all points of the field of view.

This estimate is obtained by convolution of the convolution mask corresponding to the PSF with the voxels of the space domain D_(i).

The convolution mask corresponding to the PSF of the PET scanner is obtained as follows:

Let C=(x_(c),y_(c)) and X=(x,y,z) respectively be the coordinates of the axis of the scanner and those of a voxel situated in the field of view of the scanner.

Let δX=(δx, δy, δz) be a small movement around X.

The value in X+δX of the convolution mask corresponding to the PSF in X is given by: PSF_(x,y,z)(δx,δy,δz)=ƒ(δr,PSF_(radical)(ρ),δt,PSF_(tan gential)(ρ),z,PSF_(axial)(z))  (5)

where

δr=δx cos θ+δy sin θ,

δt=−δx sin θ+δy cos θ,

cos θ=(x−x_(c))/∥X−C∥,

sin θ=(y−y_(c))/∥X−C∥,

and f is the function representing the look of the PSF.

RSF_(i)(x) is obtained by applying the convolution mask PSF_(x,y,z,)(δx,δy,δz) to the mask of D_(i). This smoothing is non-homogenous and reproduces the effects produced by the acquisition and reconstruction of the image.

In parallel, in step 22, one defines a set Λ_(i) of voxels for which the activity measured in their vicinity is representative of the tracer concentration in the space domain D_(i). These voxels can be determined automatically or manually.

S_(i) is also defined as a set of voxels to be excluded from the regions of interest R_(j).

The voxels in each space domain D_(i) are then ordered in step 24 according to the following principle.

At least one voxel xεD_(i) is affected by the effects that smooth the images (partial volume effect, fraction of tissue in the voxel, physiological movement, etc.).

It is possible to predict, with or without introducing a priori information, in which ratio this voxel x is affected by these effects from one or more predetermined order criteria.

Among these criteria:

-   -   criterion α(x): the voxel x is affected by the partial volume         effect, the tissue fraction or the periodic physiological         movements of the subject, this criterion making it possible to         define in other terms whether the activity contained in the         voxel x comes from only one or several different space domains;     -   criterion β(x): the voxel x participates in calculating         non-diagonal elements of the matrix W, this criterion being         easily calculated from the calculation of the RSF_(k)(x) for         k≠i;     -   criterion γ(x): any other criterion or set of criteria that can         help discriminate the most reliable voxels for estimating the         tracer concentration (reliability of the PET measurement,         response homogeneity within a same organ, etc.), for example         from a probabilistic atlas.

The criteria α(x), β(x) and γ(x) are defined so that the smaller they are, the lesser the aggregation of x in the region of interest R_(I) will introduce correction error due to the segmentation errors or the effects not taken into account in estimating the PSF.

Let g_(i)(α(x), β(x), γ(x), Λ_(i), S_(i)) be an order of the voxels xεD_(i) giving x a lower rank as α(x), β(x) and γ(x) are smaller, with the constraint that:

-   -   (C1) A voxel xεD_(i) of rank n is connected by lower rank voxels         to one of the points of Λ_(i); and     -   (C2) The voxels yεS_(i) are excluded from this sort order.

Hereafter, R_(i) ^((n)) will designate a ROI containing the smallest n voxels with ranks g_(i)(α(x), β(x), γ(x)) among all of the voxels of D_(i) or part of those voxels.

R_(i) ^((n)) can also contain voxels situated outside D_(i) but on the periphery thereof.

The automatic optimization of the regions of interest R_(j) is then carried out in step 26 of the inventive method using an iterative loop.

The parameters are first defined that are useful for the automatic optimization of the regions of interest R_(j).

Let R′_(i)⊂R_(i) let {circumflex over (T)}′ and {circumflex over (T)} be the estimated tracer concentrations in the organs with R′_(i) and with R_(i) respectively. Let T be the true tracer concentration. It can be put forward that:

-   -   The errors due to segmentation errors on the estimate of T by         using R′_(i) are less significant that those committed using         R_(i). Furthermore, the geometric transfer matrix W′ estimated         using R′_(i) is closer to the unit matrix than the geometric         transfer matrix W estimated using R_(i). The conditioning of the         matrix W′ is better than that of W.     -   The errors due to noise made on the estimate of T using R′_(i)         are more significant than those committed using R_(i).     -   It is possible to estimate T_(i,k,τ), which is the tracer         concentration in the region of interest R_(i) for iteration k         and measurement time τ:         -   either by optimizing R_(i) globally by incorporating the             information for all of the measurement times (hereafter             called “frames”);         -   or by optimizing R_(i) separately for each frame τ;         -   or by optimizing R_(i) separately for each frame τ_(opt)             while also integrating information coming from other frames             τ_(mes).

The last case corresponds to a general case, and the first two cases are specific cases thereof. We will therefore only consider the general case below.

R_(i,S) _(i,k) ^((n) ^(i,k) ⁾ is defined beforehand as the ROI containing in iteration k the n_(i,k) voxels having the smallest rank voxels x among all of the voxels of D_(i), excluding voxels belonging to the set S_(i,k).

Let δ be an elementary modification of the number of voxels of the regions of interest R_(i) and let Δ_(k) be a set of possible elementary modifications in iteration k.

Let

R_(i, S_(i, k, τ_(opt)))^((n_(i, k, τ_(opt)))) be the ROI defined in iteration k to optimize the region of interest R_(i) for the frame τ_(opt), containing n_(i,k,τ) _(opt) voxels and excluding the voxels from the set S_(i,k,τ) _(opt) while remaining compliant with (C1) and (C2).

Let T_(i,k,τ) _(opt) be the tracer concentration one wishes to estimate for the frame τ_(opt) and let T_(i,k,τ) _(opt) _(,τ) _(mes) be the tracer concentration estimated in

R_(i, S_(i, k, τ_(opt)))^((n_(i, k, τ_(opt)))) at time τ_(mes).

Let σ_(x,τ) _(mes) ² be the variance of the noise at the voxel x at time τ_(mes). This variance can be determined as a function of the image reconstruction algorithm and possibly the PET signal measured at voxel x at time τ_(mes).

Let t_(i,k,τ) _(opt) _(,τ) _(mes) be the average tracer concentration measured in the region of interest R_(i) at iteration k at time τ_(mes) to estimate T_(i,k,τ) _(opt) .

In certain hypotheses, the variance ζ_(i,k,τ) _(opt) _(, τ) _(mes) ² of the error due to noise made on the estimate of t_(i,k,τ) _(opt) _(, τ) _(mes) can also be determined from σ_(x,τ) _(mes) ² and from n_(i,k,τ) _(opt) .

Knowing this error, the GTM method provides an upper limit for the variance ξ_(i,k,τ) _(opt) _(, τ) _(mes) ² of the error due to noise made on the estimate of T_(i,k,τ) _(opt) :

$\begin{matrix} {\xi_{i,k,\tau_{opt},\tau_{mes}}^{2} = {\sum\limits_{i = 1}^{I}{W_{k,\tau_{opt}}^{- 1} \times \zeta_{i,k,\tau_{opt},\tau_{mes}}^{2}}}} & (6) \end{matrix}$

where W_(k,τ) _(opt) is the GTM matrix calculated in iteration k to estimate {T_(i,k,τ) _(opt) }_(1≦i≦I).

Let

h({T_(i, k, τ_(opt), τ_(mes))}_(1 ≤ τ_(mes) ≤ T, 1 ≤ i ≤ I), {ξ_(i, k, τ_(opt), τ_(mes))²}_(1 ≤ τ_(mes) ≤ T, 1 ≤ i ≤ I), {T_(i, l, τ_(opt), τ_(mes))}_(1≤ l < k, 1 ≤ τ_(mes) ≤ T, 1≤ i ≤ I), {ξ_(i, l, τ_(opt), τ_(mes))²}_(1≤ l < k, 1 ≤ τ_(mes) ≤ T, 1 ≤ i ≤ I))  of   {T_(i, k, τ_(opt), τ_(mes))}_(1 ≤ τ_(mes) ≤ T, 1 ≤ i ≤ I) be a quantitative criterion evaluating the difference related to the noise between the estimate at iteration k and the estimates of

{T_(i, l, τ_(opt), τ_(mes))}_(1≤ l < k, 1 ≤ τ_(mes) ≤ T, 1≤ i ≤ I) in the preceding iterations.

The greater the difference between

{T_(i, k, τ_(opt), τ_(mes))}_(1 ≤ τ_(mes) ≤ T, 1 ≤ i ≤ I)  and  {T_(i, l, τ_(opt), τ_(mes))}_(1≤ l < k, 1 ≤ τ_(mes) ≤ T, 1≤ i ≤ I), the smaller the values of h.

Let h_(min) be a threshold below which

{T_(i, k, τ_(opt), τ_(mes))}_(1 ≤ τ_(mes) ≤ T, 1 ≤ i ≤ I) is considered to be significantly different from

{T_(i, l, τ_(opt), τ_(mes))}_(1≤ l < k, 1 ≤ τ_(mes) ≤ T, 1≤ i ≤ I).

The algorithm of the iterative loop for optimizing regions of interest R_(j) is the following (FIG. 6).

Step 28 corresponds to the initialization of the loop.

The regions of interest R_(j) are initialized by

R_(i, S_(i, 0, τ_(opt)))^((n_(i, 0, τ_(opt)))), where n_(i,0,τ) _(opt) >2 and where S_(i,0) is an empty set.

In each iteration k, one then goes on to a step 30 for determining a set Δ_(k) of possible elementary modifications δ of the

{R_(i, S_(i, k − 1, τ_(opt)))^((n_(i, k − 1, τ_(opt))))}_(1 ≤ i ≤ I) that can result in the addition of voxels to a region of interest R_(i) and/or in the addition of voxels to S_(i) and/or in the modification of Λ_(i).

In step 32, for all δεΔ, one calculates {t_(i) ^((δ))}_(1≦i≦I), W,

{T_(i, k, τ_(opt), τ_(mes))^((δ))}_(1 ≤ τ_(mes) ≤ T, 1 ≤ i ≤ I), {ξ_(i, k, τ_(opt), τ_(mes))^(2^((δ)))}_(1 ≤ τ_(mes) ≤ T, 1 ≤ i ≤ I)  and ${h\begin{pmatrix} {\left\{ T_{i,k,\tau_{opt},\tau_{mes}}^{(\delta)} \right\}_{{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}},\left\{ \xi_{i,k,\tau_{opt},\tau_{mes}}^{2^{(\delta)}} \right\}_{{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}},} \\ {\left\{ T_{i,l,\tau_{opt},\tau_{mes}} \right\}_{{1 \leq l < k},{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}},\left\{ \xi_{i,l,\tau_{opt},\tau_{mes}}^{2} \right\}_{{1 \leq l < k},{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}}} \end{pmatrix}}.$

In step 36, the elementary modification δ_(opt) is selected that provides an estimate of the tracer concentration closest to the tracer concentrations estimated in the preceding iterations:

$\begin{matrix} {\delta_{opt} = {\underset{\delta}{{\arg\;\max}}\left( {h\begin{pmatrix} {\left\{ T_{i,k,\tau_{opt},\tau_{mes}}^{(\delta)} \right\}_{{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}},\left\{ \xi_{i,k,\tau_{opt},\tau_{mes}}^{2^{(\delta)}} \right\}_{{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}},} \\ {\left\{ T_{i,l,\tau_{opt},\tau_{mes}} \right\}_{{1 \leq l < k},{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}},\left\{ \xi_{i,l,\tau_{opt},\tau_{mes}}^{2} \right\}_{{{1 \leq l < k},{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}}\;}} \end{pmatrix}} \right)}} & (7) \end{matrix}$

In step 38, the criterion h corresponding to δ_(opt) is compared with h_(min).

If

${{h\begin{pmatrix} {\left\{ T_{i,k,\tau_{opt},\tau_{mes}}^{(\delta_{opt})} \right\}_{{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}},\left\{ \xi_{i,k,\tau_{opt},\tau_{mes}}^{2^{(\delta_{opt})}} \right\}_{{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}},} \\ {\left\{ T_{i,l,\tau_{opt},\tau_{mes}} \right\}_{{1 \leq l < k},{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}},\left\{ \xi_{i,l,\tau_{opt},\tau_{mes}}^{2} \right\}_{{1 \leq l < k},{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}}} \end{pmatrix}} > h_{\min}},$ then the elementary modification δ_(opt) is applied to R_(i) and to S_(i) (step 40) and one goes on to the following iteration k+1 by returning to step 30.

Otherwise the algorithm ends (step 42), and we have:

R_(i) = R_(i, S_(i, k − 1, τ_(opt)))^((n_(j, k − 1, τ_(opt)))); {T_(i, τ_(opt))}_(1 ≤ i ≤ I) = {T_(i, k − 1, τ_(opt), τ_(opt))}_(1 ≤ i ≤ I); and {ξ_(i, τ_(opt))²}_(1 ≤ i ≤ I) = {ξ_(i, k − 1, τ_(opt), τ_(opt))²}_(1 ≤ i ≤ I).

In reference to FIG. 3, one thus obtains, in step 44, the estimate of the true tracer concentration T_(i) in the space domains D_(i).

One also obtains, from step 26 for optimizing the regions of interest R_(j), the final size of said regions of interest (step 46) that makes it possible to optimize the parameters of the method.

Indeed, the method comprises various parameters determining the order in which the voxels will be aggregated to the ROIs. These parameters include the Λ_(i), the functions α(x), β(x), γ(x) and g_(i)(α(x), β(x), γ(x), Λ_(i), S_(i)).

The optimization method finds a middle way between estimating errors due to noise and estimating errors due to effects such as segmentation errors and physiological movements.

The errors due to noise tend to decrease as the size of the ROIs increases.

The final size of the R_(j) is therefore an indicator of:

-   -   the quality of the D_(i);     -   the homogeneity of the PET signal within the D_(i); and     -   the quality of the parameters determining the order of the         voxels.

At fixed D_(i) and homogeneity of the signal, the final size of the R_(j) is therefore an indicator of the quality of the selection of the Λ_(i), the functions α(x), β(x), γ(x) and g_(i)(α(x), β(x), γ(x), Λ_(i), S_(i)).

It is then possible to:

-   -   initiate/launch several optimizations of the R_(j) with         different D_(i), Λ_(i), α(x), β(x), γ(x) and g_(i)(α(x), β(x),         γ(x), Λ_(i), S_(i)) and choose the D_(i), Λ_(i), α(x), β(x),         γ(x) and g_(i)(α(x), β(x), γ(x), Λ_(i), S_(i)) for which the         size of the R_(j) or any increasing monotonous function of this         size or any other function quantitatively evaluating the quality         of the estimate of the T_(i) is maximal (step 48);     -   give the size of the R_(j), or the value of any monotonous         function of that size or any other function quantitatively         evaluating the quality of the estimate of the T_(i) as a global         indicator of the quality of the D_(i), Λ_(i), α(x), β(x), γ(x)         and g_(i)(α(x), β(x), γ(x), Λ_(i), S_(i)) (step 50).

One example of an embodiment of the inventive method is the following:

-   -   RSF_(i)(x) is estimated using the Frouin et al. method;     -   J=I and j=i;     -   the D_(i) are defined as the space domains defined by the         segmentation using a method based on the local mean analysis         (LMA) proposed by Maroy et al. in the article “Segmentation of         Rodent Whole-Body Dynamic PET Images: An Unsupervised Method         Based on Voxel Dynamics,” IEEE Trans. Med. Imaging, 2008;     -   Λ_(i) is the set of voxels xεD_(i) extracted during the step for         extracting local minima from the image {circumflex over (α)}_(p)         ² measuring, at each voxel p, the homogeneity of the tracer         concentration in the vicinity of p;     -   α(x) is the order of the voxels sorted by increasing {circumflex         over (α)}_(x) ²;     -   β(x) is the order of the voxels sorted by decreasing RSF_(i)(x);     -   g_(i)(α(x), β(x), γ(x), Λ_(i), S_(i)) is the order of x among         the voxels xεD_(i) from which the voxels of S_(i) are excluded         and which are sorted by increasing (a×α²(x)+(1−a)×β²(x)). In         that order, the rank of the voxels not verifying (C1) is shifted         until they can verify (C1);     -   Δ_(k)={δ_(i)}_(1≦i≦I), where δ_(i) is the elementary         modification, which consists of adding d_(i) voxels to the         region of interest R_(i);     -   τ_(opt) is the frame for which one optimizes the {R_(j)}_(1≦i≦I)         and τ_(mes) is a frame where one estimates the {T_(i}1≦i≦I);     -   let L be a scale level and L_(max) the maximum scale level, set         a priori (for example L_(max)=T/4). A set F_(L) of L+1         consecutive frames at scale level L, these L frames being         centered on τ_(opt), is associated with the frame τ_(opt) at         scale level 0;     -   let p_(i,L) be the p-value associated with the statistic t

$\mspace{20mu}{{\sum\limits_{\tau_{meas} \in F_{L}}\left( {\sum\limits_{l = 1}^{k - 1}\left( \frac{T_{i,k,\tau_{opt},\tau_{mes}}^{(\delta_{opt})} - T_{i,l,\tau_{opt},\tau_{mes}}}{\sqrt{\left( {\xi_{i,k,\tau_{opt},\tau_{mes}}^{2^{(\delta_{opt})}} + \xi_{i,l,\tau_{opt},\tau_{mes}}^{2}} \right)}} \right)} \right)};}$ ${h\begin{pmatrix} {\left\{ T_{i,k,\tau_{opt},\tau_{mes}}^{(\delta_{opt})} \right\}_{{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}},\left\{ \xi_{i,k,\tau_{opt},\tau_{mes}}^{2^{(\delta_{opt})}} \right\}_{{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}},} \\ {\left\{ T_{i,l,\tau_{opt},\tau_{mes}} \right\}_{{1 \leq \; l < k},{1 \leq \tau_{mes} \leq T},{1 \leq \; i \leq I}},\left\{ \xi_{i,l,\tau_{opt},\tau_{mes}}^{2} \right\}_{{1 \leq \; l < k},{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}}} \end{pmatrix}} = {\min\limits_{1 \leq i \leq I}{\left( {\min\limits_{L}\left( p_{i,L} \right)} \right).}}$

The algorithm is as follows.

In the initialization step:

δ_(i)=0.001×#D_(i), where #D_(i) is the cardinal of the segmented space domain D_(i), n_(i,0,τ) _(opt) =δ_(i), R_(i) is initialized by

R_(i, S_(i, 0, τ_(opt)))^((n_(i, 0, τ_(opt)))) and S_(i,0) is an empty set.

Upon each iteration k and for every i:

-   -   one calculates

{t_(i)^((δ_(i)))}_(1 ≤ i ≤ I), W, {T_(i, k, τ_(opt), τ_(mes))^((δ_(i)))}_(1 ≤ τ_(mes) ≤ T, 1 ≤ i ≤ I), {ξ_(i, k, τ_(opt), τ_(mes))^(2^((δ_(i))))}_(1 ≤ τ_(mes) ≤ T, 1 ≤ i ≤ I)  and ${h\begin{pmatrix} {\left\{ T_{i,k,\tau_{opt},\tau_{mes}}^{(\delta_{i})} \right\}_{{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}},\left\{ \xi_{i,k,\tau_{opt},\tau_{mes}}^{2^{(\delta_{i})}} \right\}_{{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}},} \\ {\left\{ T_{i,l,\tau_{opt},\tau_{mes}} \right\}_{{1 \leq \; l < k},{1 \leq \tau_{mes} \leq T},{1 \leq \; i \leq I}},\left\{ \xi_{i,l,\tau_{opt},\tau_{mes}}^{2} \right\}_{{1 \leq \; l < k},{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}}} \end{pmatrix}};{and}$ ${{let}\mspace{14mu} i} = {\quad{{{\underset{1 \leq i \leq I}{\arg\;\max}{\left( {h\left( \begin{matrix} {\left\{ T_{i,k,\tau_{opt},\tau_{mes}}^{(\delta_{i})} \right\}_{{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}},\left\{ \xi_{i,k,\tau_{opt},\tau_{mes}}^{2^{(\delta_{i})}} \right\}_{{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}},} \\ {\left\{ T_{i,l,\tau_{opt},\tau_{mes}} \right\}_{{1 \leq \; l < k},{1 \leq \tau_{mes} \leq T},{1 \leq \; i \leq I}},\left\{ \xi_{i,l,\tau_{opt},\tau_{mes}}^{2} \right\}_{{1 \leq \; l < k},{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}}} \end{matrix} \right)} \right).{If}}\mspace{14mu}{h\left( \begin{matrix} {\left\{ T_{i,k,\tau_{opt},\tau_{mes}}^{(\delta_{i})} \right\}_{{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}},\left\{ \xi_{i,k,\tau_{opt},\tau_{mes}}^{2^{(\delta_{i})}} \right\}_{{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}},} \\ {\left\{ T_{i,l,\tau_{opt},\tau_{mes}} \right\}_{{1 \leq \; l < k},{1 \leq \tau_{mes} \leq T},{1 \leq \; i \leq I}},\left\{ \xi_{i,l,\tau_{opt},\tau_{mes}}^{2} \right\}_{{1 \leq \; l < k},{1 \leq \tau_{mes} \leq T},{1 \leq i \leq I}}} \end{matrix} \right)}} > h_{\min}},}}$ then d_(i) voxels are added to n_(i,k,τ) _(opt) ,

R_(i, S_(i, k, τ_(opt)))^((n_(i, k, τ_(opt)))) is redetermined and one moves on to the following iteration k+1.

Otherwise, if d_(i)>1, d_(i) is decreased and another iteration is tried.

Otherwise, if #S_(i)<s_(max), the d_(i) above voxels are added to S_(i), d_(i) is increased and another iteration is tried.

Otherwise, the algorithm is stopped and the solution is:

{n_(i, τ_(opt))}_(1 ≤ i ≤ I) = {n_(i, k − 1, τ_(opt))}_(1 ≤ i ≤ I); R_(i) = R_(i, S_(i, k − 1, τ_(opt)))^((n_(i, k − 1, τ_(opt)))); {T_(i, τ_(opt))}_(1 ≤ i ≤ I) = {T_(i, k − 1, τ_(opt), τ_(opt))}_(1 ≤ i ≤ I):  and {ξ_(i, τ_(opt))²}_(1 ≤ i ≤ I) = {ξ_(i, k − 1, τ_(opt), τ_(opt))²}_(1 ≤ i ≤ I).

The results were obtained with the following material.

For the simulation, we simulated, using an analytical simulator, fifty PET acquisitions of the MOBY mouse phantom on a PET scanner dedicated to small animals (the simulated scanner was the FOCUS 220 by Siemens with a spatial resolution of 1.3 mm), with time frames of one minute each. The images were reconstructed using the OSEM (Ordered Subsets Expectation Maximization) method, which is a traditional PET image reconstruction method, with voxels of 0.5×0.5×0.8 mm³.

The experimental data include sixteen mice injected with human peritoneal tumor xenografts. The mice received an injection of Fluoro-thymidine (18F-FLT), immediately followed by a PET acquisition of 5400 min made up of 18 time frames (five one-minute frames, five two-minute frames, three five-minute frames, three ten-minute frames, and two fifteen-minute frames). The images were reconstructed using the OSEM method with voxels of 0.5×0.5×0.8 mm³. The mice were sacrificed immediately after the acquisition, and the tracer concentration was counted in the sampled organs. This concentration counted in the organs served as a “gold standard” for the concentration estimated at the end of the acquisition.

The PET images were segmented using the LMA method (FIG. 7A for the simulation and FIG. 7B for the experimental data).

The estimation of the tracer concentration in the organs is done using three methods:

-   -   calculating the average TAC within the segmented space domain         corresponding to the organ;     -   partial volume correction carried out via the GTM method         implemented in the image space as described by Frouin et al.         (simply called “GTM” hereafter); and     -   partial volume correction carried out using the method proposed         in the example embodiment of the present invention (called         “LMA-GTM” hereafter).

The GTM and LMA-GTM methods were compared based on:

-   -   ETR: which is the absolute value deviation between the         corresponding measurement of the gold standard and the measured         value, expressed in percentage of the gold standard measurement;         and     -   ARC: which is the apparent recovery coefficient, in other words,         the measured value divided by the corresponding value of the         gold standard.

FIGS. 8A and 8B show examples of time variation of the tracer concentration (TACs) estimated for the simulations (FIG. 8A) and for the experimental data (8B).

FIGS. 9A and 9B show that the LMA-GTA method significantly improves the precision of the measurements (the p-value is less than 10⁻⁵ both for the simulations and the experimental data) and that the GTM method only improves the TACs extracted for only the experimental data.

The precision of measurements estimated using the LMA-GTM method is good (ETRs_(Simulations)=5.3%±8% and ETR_(Données expérimentales)=4.8%±7%) and the apparent contrast is correctly recovered (ARC_(Simulations)=94%±10% and ARC_(Données expérimentales)=99.98%±9%).

The results obtained using the LMA-GTM method are significantly better in terms of ARC and ETR than those obtained using the GTM method (the p-value is less than 10⁻⁵).

The complete process of extracting corrected TAC from the partial volume takes 10 minutes per image:

-   -   segmentation (˜15 seconds);     -   naming of the organs by the user (˜5 minutes); and     -   partial volume correction (˜4 minutes).

FIG. 10A shows the ARC obtained by the three methods on the simulation images for each organ.

For the LMA-GTM method, the ARC varies between 94%±5% and 99.3%±1% for all of the organs with the exception of the small organs, such as the thalamus and the thyroid, and the pancreas due to its shape.

The ETR (FIG. 10B) is below 10% of the true value for all of the organs except the thalamus (ETR=32%±5%) and the thyroid (ETR=21%±6%).

The precision obtained using the LMA-GTM method is better than that obtained by calculating the average TAC and using the GTM method, and with a p-value below 10⁻⁵ for all of the organs.

The results obtained for the experimental data are similar to those obtained for the simulations.

The contrast recovery (FIG. 11A) was good (between 98.4%±7% and 110%±6%).

Furthermore, the ETR (FIG. 11B) was below 10% for the LMA-GTM method for all of the organs.

In both cases, the LMA-GTM method obtains better results than the GTM method (p-value less than 0.002).

For brain studies, the GTM method is considered one or even the reference method for correcting the partial volume effect in PET images.

For studies in rodents, to our knowledge there is no generic method (applicable to all tracers and all pathologies) for partial volume correction.

The LMA-GTM method shows strong potential for such studies, but also probably for studies in humans, in whole-body or brain images.

Indeed, the LMA-GTM method:

-   -   uses the LMA automatic definition of organs in the PET images;     -   does not require anatomical or a priori information;     -   is robust to segmentation errors and errors due to effects that         cannot be modeled by the PVE correction; and     -   is fast, easy to use, and precise.

The present invention therefore proposes a method making it possible to reliably and precisely estimate the concentration of a tracer in a tissue structure assembly by optimally defining the regions of interest in which the tracer concentration is measured. 

The invention claimed is:
 1. A method for estimating the concentration of a tracer in a tissue structure assembly including at least one tissue structure, from a measurement image of the tracer concentration in said tissue structure assembly, which is obtained by an imaging apparatus, the image comprising at least one space domain (D_(i)) inside which the tracer concentration is homogenous and at least one region of interest (R_(i)) inside which the tracer concentration is measured, the method comprising: determining a geometric transfer matrix (W) having coefficients (w_(i,j)) representative of the contribution of the space domains (D_(i)) in the measurement of the tracer concentration in the regions of interest (R_(j)); optimizing the coefficients (w_(i,j)) of the geometric transfer matrix (W) by defining the best regions of interest (R_(j)) in terms of errors in order to measure the tracer concentration, the definition of the regions of interest (R_(j)) being carried out according to an iterative loop that includes the following steps upon each iteration (k): modifying the regions of interest (R_(j)); and calculating the coefficients (w_(i,j)) of the geometric transfer matrix (W) from the modified regions of interest (R_(j)); selecting an optimized geometric transfer matrix (W) among the calculated geometric transfer matrices (W); and estimating the tracer concentration from the optimized geometric transfer matrix (W).
 2. The method according to claim 1, wherein the modifying the regions of interest (R_(j)) comprises the addition and/or exclusion of at least one image element (x).
 3. The method according to claim 1, wherein the iterative loop for defining the regions of interest (R_(j)) comprises, for each iteration (k): determining a set of elementary modifications (δ) each including adding at least one image element (x) to the regions of interest (R_(j)); estimating, for each elementary modification (s), the tracer concentration in the regions of interest (R_(j)) modified by the elementary modification (s); evaluating, for each elementary modification (s), a criterion (h) making it possible to compare the estimated tracer concentration with the tracer concentrations estimated in at least one preceding iteration; selecting the elementary modification (δ_(opt)) yielding a tracer concentration estimate closest to the tracer concentrations estimated in the preceding iterations; comparing the criterion (h) corresponding to the selected elementary modification (δ_(opt)) with a predetermined threshold (h_(min)); and selectively applying or not applying the selected elementary modification (δ_(opt)) or stopping the iterative loop as a function of the result of the comparison of the criterion (h) with the threshold (h_(min)).
 4. The method according to claim 1, further comprising ordering image elements (x) in each space domain (D_(i)) as a function of order criteria making it possible to evaluate a risk of introducing errors not due to noise into the tracer concentration estimate.
 5. The method according to claim 4, wherein the order criteria comprise at least one of (a) a first criterion (α(x)) making it possible to define whether the concentration contained in the image element (x) comes from one or more different spatial domains (D_(i)), (b) a second criterion (β(x)) making it possible to define whether the image element (x) participates in the calculation of the non-diagonal elements of the geometric transfer matrix (W), and (c) a third criterion (γ(x)) making it possible to define whether the image element (x) is reliable for estimating the tracer concentration.
 6. The method according to claim 5, wherein the first criterion (α(x)) makes it possible to measure the homogeneity of the tracer concentration near the image element (x), the second criterion (β(x)) makes it possible to measure the contribution of the space domains (D_(i)) to the measurement of the tracer concentration in the image element (x), and the third criterion (γ(x)) is obtained from a probabilistic atlas.
 7. The method according to claim 4, wherein the iterative loop for defining the regions of interest (R_(j)) is carried out following the order of the image elements (x) obtained in the ordering step.
 8. The method according to claim 1, further comprising estimating a point spread function (PSF) of the imaging apparatus at any point of the imaging apparatus's field of view.
 9. The method according to claim 8, wherein the estimating the point spread function (PSF) comprises: measuring the point spread function (PSF) at several points of the field of view; and interpolating and/or extrapolating the obtained measurements.
 10. The method according to claim 8, further comprising estimating a region spread function (RSF) for each space domain (D_(i)) by convolution of the point spread function (PSF) with each space domain (D_(i)).
 11. The method according to claim 1, further comprising calculating the size of the regions of interest (R_(j)).
 12. The method according to claim 11, further comprising delimiting space domains (D_(i)) so as to obtain regions of interest (R_(i)) with a maximum size.
 13. The method according to claim 11, further comprising: ordering image elements (x) in each space domain (D_(i)) as a function of order criteria making it possible to evaluate a risk of introducing errors not due to noise into the tracer concentration estimate; and optimizing the order criteria so as to obtain regions of interest (R_(j)) with a maximum size.
 14. The method according to claim 1, wherein the measurement image of the tracer concentration in the tissue structure is a sequence of matched three-dimensional images and includes at least one three-dimensional image, and the image elements (x) are voxels.
 15. The method according to claim 14, wherein the at least one three-dimensional image is obtained by positron emission tomography.
 16. A non-transitory computer-readable medium including a computer-executable code to estimate the concentration of a tracer in a tissue structure assembly including at least one tissue structure from a measurement image of the tracer concentration in the tissue structure assembly obtained by an imaging apparatus, the image including at least one space domain (D_(i)) inside which the tracer concentration is homogenous and at least one region of interest (R_(i)) inside which the tracer concentration is measured, the code comprising computer-executable instructions to perform a method when the instructions are executed by a data processing system, the method comprising: determining a geometric transfer matrix (W) having coefficients (w_(i,j)) representative of the contribution of the space domains (D_(i)) in the measurement of the tracer concentration in the regions of interest (R_(j)); optimizing the coefficients (w_(i,j)) of the geometric transfer matrix (W) by defining the best regions of interest (R_(j)) in terms of errors in order to measure the tracer concentration, the definition of the regions of interest (R_(j)) being carried out according to an iterative loop that includes the following steps upon each iteration (k): modifying the regions of interest (R_(j)); and calculating the coefficients (w_(i,j)) of the geometric transfer matrix (W) from the modified regions of interest (R_(j)); selecting an optimized geometric transfer matrix (W) among the calculated geometric transfer matrices (W); and estimating the tracer concentration from the optimized geometric transfer matrix (W).
 17. A device configured to estimate the concentration of a tracer in a tissue structure assembly including at least one tissue structure from a measurement image of the tracer concentration in the tissue structure assembly obtained by an imaging apparatus, the image including at least one space domain (D_(i)) inside which the tracer concentration is homogenous and at least one region of interest (R_(j)) inside which the tracer concentration is measured, the device comprising: the imaging apparatus; and a data processing system programmed to: determine a geometric transfer matrix (W) having coefficients (w_(i,j)) representative of the contribution of the space domains (D_(i)) in the measurement of the tracer concentration in the regions of interest (R_(j)); optimize the coefficients (w_(i,j)) of the geometric transfer matrix (W) by defining the best regions of interest (R_(j)) in terms of errors in order to measure the tracer concentration, the definition of the regions of interest (R_(j)) being carried out according to an iterative loop that includes the following steps upon each iteration (k): modifying the regions of interest (R_(j)); and calculating the coefficients (w_(i,j)) of the geometric transfer matrix (W) from the modified regions of interest (R_(j)); select an optimized geometric transfer matrix (W) among the calculated geometric transfer matrices (W); and estimate the tracer concentration from the optimized geometric transfer matrix (W). 